Electronics/Electronics Formulas/Series Circuits/Series LC

Circuit Configuration

Formula

The total Impedance of the circuit

$Z=Z_{R}+Z_{L}$

$Z=R+j\omega L$

$Z={\frac {1}{R}}(1+j\omega T)$

$T={\frac {L}{R}}$

The Differential equation of the circuit at equilibrium

$L{\frac {di}{dt}}+{\frac {1}{C}}\int idt=0$

${\frac {d^{2}i}{dt^{2}}}+{\frac {1}{LC}}=0$

$s^{2}+{\frac {1}{LC}}=0$

$s=\pm j{\sqrt {\frac {1}{LC}}}t$

$s=\pm j\omega t$

The Natural Response of the circuit

$i=ASin\omega t$

The Resonance Response of the circuit

$Z_{L}-Z_{C}=0$ . $Z_{L}=Z_{C}$ . $\omega L={\frac {1}{\omega C}}$ . $\omega ={\sqrt {\frac {1}{LC}}}$

$V_{L}+V_{C}=0$ . $V_{C}=-V_{L}$

Summary

Series LC can be characterised by

2nd order Differential Equation

${\frac {d^{2}i}{dt^{2}}}+{\frac {1}{T}}=0$

$T=LC$

With Natural Response of a Wave function

$i=ASin\omega t$

With Resonance Response of a Standing Wave function

$i=ASin\omega t$